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Proposition: A General Criterion for the Convergence of Infinite Complex Series
A complex infinite series \(\sum_{k=0}^\infty x_k\) is a convergent complex series, if and only if for every \(\epsilon > 0\) there is an index \(N(\epsilon)\in\mathbb N\) such that
\[\left\sum_{k=m}^n x_k\right < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).\]
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983